LaPlace Transform Pairs

In mathematics, the
LaPlace transform is one of the best known and most widely used integral transforms. It is commonly used to
produce an easily solvable algebraic equation from an ordinary differential equation. It has many important
applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and
probability theory.

In mathematics [and engineering], it is used for solving differential and integral
equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical
circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform
is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time,
to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians
per unit time. Given a simple mathematical or functional description of an input or output to a system, the
Laplace transform provides an alternative functional description that often simplifies the process of analyzing
the behavior of the system, or in synthesizing a new system based on a set of specifications.
- Wikipedia